“微积分二：数列与级数”将介绍数列、无穷级数、收敛判别法和泰勒级数。本课程不仅仅满足于得到答案，而且要做到知其然，并知其所以然。

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微积分二: 数列与级数 (中文版)

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“微积分二：数列与级数”将介绍数列、无穷级数、收敛判别法和泰勒级数。本课程不仅仅满足于得到答案，而且要做到知其然，并知其所以然。

From the lesson

幂级数

在第五个模块中，我们学习幂级数。截至目前为止，我们一次讲解了一种级数；对于幂级数，我们将讲解整个系列取决于参数 x 的级数。它们类似于多项式，因此易于处理。而且，我们关注的许多函数，如 e^x，也可表示为幂级数，因此幂级数将轻松的多项式环境带入棘手的函数域，如 e^x。

- Jim Fowler, PhDProfessor

Mathematics

Half open intervals.

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Well as an example, let's consider this series.

The sum n goes from one to infinity of x to the n over n.

Well it's a power series, so what's its interval of convergence?

We'll fully address that question if you asked the easier question.

What 's its radius of convergence?

Well, apply the ratio test.

So, I'm going to think about where this series converges absolutely.

For which values of x is it converging absolutely?

Supposing when does this converge.

The ratio test tells me to look at this limit.

The limit as n approaches infinity of the n plus first term,

which is x to the n +1 over n +1,

divided by the nth term, which is x to the n over n.

And looking at this with the absolute value bars,

it's actually absolute convergence.

So the ratio test asks me to consider, for

which values of x is this limit less than 1?

Well I can simplify this limit somewhat.

All right, this is the limit as n approaches infinity.

Again, x to the n +1 over x to the n here, those cancel and just leave me with an x.

And here I've got n +1 in the denominator of the numerator, and

an n in the denominator of the denominator.

This ends up being, n over n + 1.

So now, whats this limit?

Well the limit of n over n + 1, as n approaches infinity,

that's 1 and this is just a constant.

So this limit of the constant, as far as n is concerned, so

this limit is just the absolute value of x.

So this series converges absolutely when the absolute value of x is less than 1.

And it diverges when the absolute value of x is bigger than 1.

So what is the radius of convergence?

Well, this is telling me that the radius of convergence is 1.

So I could plot the points on the number line where this series converges.

And what I know thus far is that it converges when the absolute

value of X is less than 1, meaning that X is between -1 and 1,

and it diverges when the absolute value of X is bigger than 1.

So I know it diverges out here and it diverges down here.

What about the end points?

Well exactly.

What happens when X is equal to 1?

Does the series converge or diverge at 1?

What happens when X is equal to -1?

Does the series converge or diverge at -1?

All I know so far is that it converges absolutely between -1 and 1, and

it diverges out here.

But I haven't actually addressed the question of whether or

not this series converges at the end points.

Well we can plug in X = 1 and then recognize the series.

Yeah, so if we're looking at this series, the sum n goes from 1 to

infinity of x to the n over n.X And I plug in x equals 1.

What do I get?

Well then this is just the sum, n goes from 1 to infinity of 1 to the n,

which is 1, over n.

Does this series converge or diverge?

That's the harmonic series.

And the harmonic series diverges,

which means that this series, when x = 1, diverges.

Now what about x = -1?

Well in that case, I'm going to plug in -1 here, and

I'll get the sum n goes from 1 to infinity of -1 to the n over n.

And that's the alternating harmonic series and that converges, albeit conditionally.

We can summarize this.

So, putting this all together, we can write the following.

We can say that the interval of convergence

is the half-open interval -1 to 1,

but closed on the -1 side because this series

converges here but doesn't converge at 1.

So the interval of convergence is this half open interval.

And note just how complicated this was.

We've got our interval of convergence and in the interior of that interval,

we've got absolute convergence.

So it wasn't too hard to figure out the radius of that interval.

But the story became way more complicated at the end points.

We had one endpoint where the series converge and

another end point where the series diverge.

And in general this is how it's going to work out.

It won't be hard for you to find the radius of convergence, but

it might be really painful to analyze the story at the end points.

And it is possible that the series could diverge at both end points,

it might converge at both end points, it might just converge at one end point.

Story at the end points is more complicated.

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